5. given: ∠5≅∠16;∠2≅∠4 prove: ∠11 and ∠14 are supplementary statements reasons

Question

Accepted Answer

# Explanation:
## Step1: Recall angle - congruence properties
We know that vertical angles are congruent. Corresponding - angles formed by parallel lines are congruent and supplementary angles add up to 180 degrees.
## Step2: Use given angle - congruence
Since $\angle5\cong\angle16$ and $\angle2\cong\angle4$, we can start building our proof using angle - relationship rules.
## Step3: Analyze supplementary angles
Given that $\angle11$ and $\angle14$ are supplementary ($\angle11+\angle14 = 180^{\circ}$).
## Step4: Apply parallel - line and angle - congruence rules
If we assume some lines are parallel (not explicitly stated in the problem - statement but can be inferred from the angle - congruence and supplementary - angle relationships), we can use the following:
   - If two lines are cut by a transversal, and corresponding angles are congruent, the lines are parallel.
   - If two lines are parallel, then same - side interior angles are supplementary.
Let's assume we are trying to prove some lines are parallel.
   - Since $\angle5\cong\angle16$, if they are corresponding angles formed by two lines and a transversal, then those two lines are parallel.
   - Since $\angle2\cong\angle4$, if they are corresponding angles, it can be related to the parallel - line situation.
   - For $\angle11$ and $\angle14$ being supplementary, if they are same - side interior angles, we can prove line - parallelism.
Let's fill in the two - column proof:
| Statements | Reasons |
|--|--|
| 1. $\angle5\cong\angle16$ | Given |
| 2. $\angle2\cong\angle4$ | Given |
| 3. $\angle11+\angle14 = 180^{\circ}$ | Given |
| 4. Assume lines $AB$ and $CD$ are cut by transversal $EF$ (not shown in the problem - figure but a general setup for angle - relationships) | For the sake of proof - construction |
| 5. If $\angle5$ and $\angle16$ are corresponding angles, then the lines they are associated with are parallel (say $l_1\parallel l_2$) | Corresponding angles postulate (if corresponding angles are congruent, then the two lines are parallel) |
| 6. If $\angle2$ and $\angle4$ are corresponding angles, it supports the parallel - line relationship | Corresponding angles postulate |
| 7. Since $\angle11$ and $\angle14$ are supplementary and are same - side interior angles (assuming the correct line - transversal setup), the lines are parallel | Same - side interior angles postulate (if same - side interior angles are supplementary, then the two lines are parallel) |

However, without a clear statement of what we are exactly proving (e.g., proving two lines are parallel), we can't complete the proof fully. But if we assume we want to prove some lines are parallel based on the given angle - congruence and supplementary - angle information:
# Answer:
The proof steps involve using the given angle - congruence ($\angle5\cong\angle16$, $\angle2\cong\angle4$) and supplementary - angle ($\angle11+\angle14 = 180^{\circ}$) information along with angle - postulates such as the corresponding - angles postulate and the same - side interior angles postulate to prove line - parallelism. The full two - column proof depends on the exact geometric statement we are trying to prove (e.g., if we are proving two lines are parallel). The key statements and reasons are as shown in the table above.

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QUESTION IMAGE

Question

Explanation:

Step1: Recall angle - congruence properties

Step2: Use given angle - congruence

Step3: Analyze supplementary angles

Step4: Apply parallel - line and angle - congruence rules

Answer:

Statements	Reasons
2. \(\angle2\cong\angle4\)	Given
3. \(\angle11+\angle14 = 180^{\circ}\)	Given
4. Assume lines \(AB\) and \(CD\) are cut by transversal \(EF\) (not shown in the problem - figure but a general setup for angle - relationships)	For the sake of proof - construction
5. If \(\angle5\) and \(\angle16\) are corresponding angles, then the lines they are associated with are parallel (say \(l_1\parallel l_2\))	Corresponding angles postulate (if corresponding angles are congruent, then the two lines are parallel)
6. If \(\angle2\) and \(\angle4\) are corresponding angles, it supports the parallel - line relationship	Corresponding angles postulate
7. Since \(\angle11\) and \(\angle14\) are supplementary and are same - side interior angles (assuming the correct line - transversal setup), the lines are parallel	Same - side interior angles postulate (if same - side interior angles are supplementary, then the two lines are parallel)